You downloaded paypal? That makes as much sense as downloading your bank. It is an account, not an app. I am sure they have an app, but you do not need an app, you need an account. After that, when you pay with paypal online, you enter your mail adress and password and paypal charges you by whatever means you told paypal to use. (And as far as I remember, they do check if your payment options work by this strange low level security thing of charging you random amounts and you needing to enter those cent amounts as password. Or something similar. Maybe I misremember.)
The process for "debit cards" is the same as for a credit card. You need the 16 digit number, the code on the back and the expiration date.
Were your previous payments with the card like this? Or did you pay by other means with the associated banking account?